The problem asks to determine the causal signal $x(n)$ given the z-transform $X(z)$, which is expressed as:

$X(z) = \frac{1}{(1 + z^{-1})(1 - z^{-1})^2}$

The solution involves expanding $X(z)$ into terms of positive powers of $z$, which is a necessary step to recover the causal signal.

Step 1: Rewriting $X(z)$

First, we rewrite $X(z)$ in terms of positive powers of $z$. The given transform is:

$X(z) = \frac{1}{(1 + z^{-1})(1 - z^{-1})^2}$

Multiplying both the numerator and denominator by $z^3$ (to eliminate negative powers of $z$):

$X(z) = \frac{z^3}{(z+1)(z-1)^2}$

Now, $X(z)$ is expressed as:

$X(z) = \frac{z^3}{(z+1)(z-1)^2}$

Step 2: Partial Fraction Expansion

Next, we expand this rational function into partial fractions:

$\frac{z^3}{(z+1)(z-1)^2} = \frac{A_1}{z+1} + \frac{A_2}{z-1} + \frac{A_3}{(z-1)^2}$

Where $A_1$, $A_2$, and $A_3$ are constants that we will solve for.

Step 3: Solving for Coefficients

• $A_1$: To find $A_1$, multiply both sides of the partial fraction expansion by $(z+1)$ and evaluate at $z = -1$:

$A_1 = (z+1)^2 F(z) \Big|_{z=-1} = \frac{z^3}{(z-1)^2} \Big|_{z=-1} = \frac{(-1)^3}{((-1)-1)^2} = \frac{-1}{4} = \frac{1}{4}$

• $A_3$: To find $A_3$, multiply both sides by $(z-1)^2$ and evaluate at $z = 1$:

$A_3 = (z-1)^2 F(z) \Big|_{z=1} = \frac{z^3}{z+1} \Big|_{z=1} = \frac{1^3}{1+1} = \frac{1}{2}$

• $A_2$: To find $A_2$, differentiate the equation and solve at $z = 1$: